1. Field of the Invention
The field of the invention is analysis of a stream of signal values by pattern detection and selection, and in particular, the display and analysis of electroencephalic signal values by mapping, pattern identification, and pattern confinement.
2. Description of Related Art
A great body of evidence suggests that electromagnetic signals recorded from the human head reflect brain function and other physiologic processes. Recording and analysis of such signals have been conducted by computer because of the high speed of brain events, the electromagnetic character of the values measured, and the great volume of signal values. It is essential to segregate the signal values detected by electrodes on the human scalp into subsets because a wide variety of sources contribute to the signal values, including multiple sources from within the brain, as well as signals from muscles such as those in the head and neck. In addition, there may also be contributions to the signals from external sources such as the computer itself, other electric appliances, and electric wiring.
Typically in current practice, a variety of electronic filters and mathematical averaging methods have been employed to pre-process the computer-recorded stream of signal values to eliminate “noise” and “artifact”. Signal values which fell too far outside rather arbitrary limits were rejected as “artifacts”, which were viewed as “distorting” the signal. The “noise” was viewed as impossible to analyze and thus assumed to have no meaning for the simple reason that no means of interpreting it was known. Thus, an initial subset of signal values was obtained by “filtering out” and discarding “noise” and “artifact”, leaving for analysis those values which were presumed to represent the “true” signal.
The pre-processing done in current practice imposes a conceptual paradigm upon the stream of signal values which then shapes all subsequent analysis of the signal values. A wave function is the dominant historic conceptual paradigm according to which electromagnetic signals recorded from the brain have been “transformed” or “inverted”. The resulting analytic display of the transformed stream of signal values is referred to in common parlance as “brain waves”. Typically the analytic wave function of choice was the sine wave employed in a Fourier Transform and implemented on a digital computer as a Fast Fourier Transform or FFT, though sometimes a Mellin or “double” Fourier Transform was used. Due to well-recognized limitations of the Fourier Transform and “wave” paradigms, new conceptual paradigms have been sought to analytically transform computer-recorded streams of signal values which can be presented as meaningfully segregated subsets. More recently, wavelet transforms have been employed with a wide variety of signals, including electrophysiologic signals as well as sound and other vibrational signal forms.
Ideally, a conceptual paradigm should enable mathematically-reversible transformations which index a stream of signal values into meaningful subsets from which one can infer a causal relation to body function or to external stimuli, and thus decompose the stream of signal values into subsets, or component signal ‘events’, for which a causal relationship can be empirically sought. The difficulty of indexing a raw stream of signal values into meaningful subsets is referred to as the “inverse problem”.
Applicants' U.S. Pat. No. 5,218,530 applies modern mathematics of chaos, non-linear dynamics and phase space portraits to display a stream of raw signal values from scalp electrodes for the purpose of visually detecting patterns in the stream of signal values. It then employs computer-implemented selection sets to segregate the subsets of signal values which contribute to the visually identified pattern. Some examples of implementation of “selection setst” within drawings displayed on a computer monitor is exemplified by AutoCAD Reference Manual, July 1986, copyright Autodesk, Inc., Sec. 2.10, pp. 45-49. AutoCAD is a registered trademark of Autodesk, Inc.
The instant invention employs the ‘chaos game’ for contraction mapping of streams of signal values to deterministic fractals which are defined as the attractors of Iterated Function Systems. In principle, the method could also be adapted to probabilistic Iterated Function Systems, though the present implementation does not do so. The mapped signal values then are searched for patterns. When a pattern is detected is confined, or encompassed, by an exclusive “selection set”. The confined subset of signal values then automatically is segregated from the stream of signal values by computer.
Some background of art related to the ‘chaos game’ and to contraction mapping to deterministic fractals with Iterated Function Systems follows.
Modern mathematics of topology and fractals are explained in Fractals for the Classroom, by Heinz-Otto Peitgen, Hartmut Jurgens, and Dietmar Saupe, published by Springer-Verlag, New York, 1992, ISBN 0-387-97041-X. Deterministic fractals called the Sierpinski gasket and the Sierpinski carpet are defined, id., at chapter 2.2, pp. 91-95, and methods of indexing them with addresses are defined id., at chapter 2.7, pp. 128-127; chapter 2.10, pp. 148-150; chapter 6.2, pp. 330-338. The Menger sponge deterministic fractal is defined, id., at p. 124 and 131-132. More detailed application of the “chaos game” and use of probabilities in its implementation in computer programming, are discussed in Peitgen, H.-O.; Jurgens, H.; and Saupe, D; Chaos and Fractals, New Frontiers of Science, Springer-Verlag, New York, 1992, Chap. 6, pp. 297-352.
The terms “deterministic fractal” and “Iterated Function System” sometimes are used nearly interchangeably. It should be understood, however, that an Iterated Function System is a process, usually though not always implemented on a computer, to approximate a mathematically precise object called a deterministic fractal. The deterministic fractal is the theoretical limit object resulting from an infinite number of iterations of the Iterated-Function System. The “deterministic fractal” therefore is called the “attractor” of the iterative process called the “Iterated Function System”.
An algorithm called the ‘chaos game’ for generating the Sierpinski gasket with random numbers is defined, id., at chapter 1.3 at pp. 41-43. More general use of the ‘chaos game’, using random numbers to map a variety of fractal forms as attractors of Iterated Function Systems (“IFS”), is explained, id., at chapter 6.2, p. 339-344. Methods of defining and generating Iterated Function Systems are described, id., at Chapter 5, pp. 255-318. The “contraction mapping principle” is explained, id. at Chapter 5.5, pp. 284-293, and especially at p. 287. Deterministic fractals such as the Sierpinski gasket, Sierpinski carpet and Menger sponge can be viewed as attractors of Iterated Function Systems. In addition, these attractors are universal indices for certain topologic classes of objects. Id., at Chapter 2.7, pp. 128-137. This means, in a topologic sense, that these attractors contain, or index, within themselves an enormous collection of mathematically distinct, but less complex, attractors. Id. That is, the attractors of these deterministic fractals are themselves constructed of less-complex attractors.
The Mandelbrot set, M, may be viewed as an attractor of an Iterated Function System. The Mandelbrot set also may be viewed as a universal index of topologic entities within the mathematical class called Julia Sets. Julia sets are defined, id., at Chapter 2.8, pp. 138-141. The Mandelbrot set, and methods of generating it by computer iteration, are defined in The Science of Fractal Images, by Heinz-Otto Peitgen and Dietmar Saupe, eds., Springer-Verlag, New York, 1988, ISBN 0-387-96608-0 and ISBN 3-540-96608-0, at chapter 4.2.1, pp. 177-179.
Analytical addressing of points mapped to the attractor of an Iterated Function System using the chaos game is described in Peitgen, H.-O.; Jurgens, H.; Saupe, D., Chaos and Fractals, New Frontiers of Science, Springer-Verlag, 1992, Chap. 6, p. 297, et seq. See also, Barnsley, M. F., Fractal Modelling of Real World Images, in Peitgen, H.-O; Saupe, D. The Science of Fractal Images, Springer-Verlag, New York, 1988.
A notable difference between the Sierpinski and Menger fractal objects, on the one hand, is that they are self-similar by linear contraction mapping, whereas Julia sets and the Mandelbrot set, on the other hand, are self-similar by nonlinear contraction mapping. Fractals for the Classroom, Chapter 2.8, p. 141.
Contraction mapping using Iterated Function Systems has been employed for image compression. Barnsley, U.S. Pat. No. 5,065,447. See also Barnsley, Michael F.; Hurd, Lyman P., Fractal Image Compression, A K Peters, LTD, Wellesley, Mass., 1993; Barnsley, Michael P., Fractals Everywhere, 2nd. Ed., Academic Press Professional, Boston, 1993 (also first ed. 1988). The mathematics of incomplete images, or the addresses of missing pieces, of the mapped fractal are discussed in Fractals Everywhere, pp. 122-125, and FIG. IV.100.
It is known that if the numbers employed in the ‘chaos game’ are not random then incomplete forms of the deterministic fractals will be generated by the ‘chaos game’ as explained in Fractals for the Classroom, Chapters 6.3 and 6.4, pp. 345-364.
Both deterministic and probabilistic Iterated Function Systems can be implemented on neural networks to generate more quickly images of fractal objects. Stark, Jaroslav; “Iterated Function Systems as Neural Networks”, Neural Networks, Vol. 4., pp. 679-690, 1991 (received 26 Jan. 1990; revised and accepted 20 Feb. 1991).